Which artificial intelligence algorithms could use tensor specific hardware?
There are a lot of other potential applications. It's a good idea to start with GPU related problems, since GPUs are essentially doing a slightly wider set of operations, slightly slower. Some possible problems where TPUs might be advantageous are:  


Shaders are algorithms for rendering graphics in one style or another. Since computer graphics can be understood as mostly linear algebra, it is natural to view this as operations over tensors.
Physical simulations, which again involve multiplication of vectors by a series of matrices. 
Options Pricing, which again involves the multiplication of vectors by a series of matrices, especially when more complex derivatives are prices and the lattice becomes multi-dimensional. 


Within AI, there are many other algorithms optimized to work with GPUs, and that could be modified to work with tensor specific hardware. For example, we have:


Genetic Programming on GPUs
Multiagent Systems on GPUs
SAT solving on GPUs


There does not yet seem to be much work optimizing these other problems for Tensor Processing units, but TPUs are also not yet very old. It took several years following the availability of inexpensive consumer GPUs before we started to see widespread use in AI. I suspect we will see more TPU-tailored code for other problems soon.